%I #27 Feb 28 2021 20:48:33
%S 47,61,73,103,137,157,167,179,223,257,263,337,347,383,467,563,613,719,
%T 733,757,769,877,887,1021,1097,1223,1297,1327,1367,1453,1481,1571,
%U 1613,1621,1759,1811,1987,1997,2003,2027,2039,2129,2251,2473,2477,2539,2593,2633,2767,2879,3001,3037,3083,3119
%N Primes that can be represented as (1/2)*Sum_{i=0..m} binomial(m,i)*prime(i+k) for some k and m >= 2.
%C Each prime is just listed once, though it may arise in more than one way.
%H Robert Israel, <a href="/A342093/b342093.txt">Table of n, a(n) for n = 1..10000</a>
%e a(3) = 73 is a term because it is prime and Sum_{i=0..2} binomial(2,i)*prime(11+i) = 31+2*37+41 = 2*73.
%e a(10) = 257 arises in two ways:
%e 2*257 = Sum_{i=0..3} binomial(3,i)*prime(17+i) = 59+3*61+3*67+71
%e and Sum_{i=0..4} binomial(4,i)*prime(9+i) = 23+4*29+6*31+4*37+41.
%p N:= 10^4: # for terms <= N
%p S:= {}:
%p for m from 2 do
%p for k from 2 do
%p v:= add(binomial(m,i)*ithprime(i+k),i=0..m)/2;
%p if v > N then break fi;
%p if isprime(v) then
%p S:= S union {v}; count:= count+1;
%p fi;
%p od;
%p if k = 2 then break fi
%p od:
%p sort(convert(S,list));
%Y Contains A338273.
%K nonn
%O 1,1
%A _J. M. Bergot_ and _Robert Israel_, Feb 28 2021
